Universal Properties of Two-Dimensional Boson Droplets

We consider a system of $N$ nonrelativistic bosons in two dimensions, interacting weakly via a short-range attractive potential. We show that for $N$ large, but below some critical value, the properties of the $N$-boson bound state are universal. In particular, the ratio of the binding energies of $(N+1)$- and $N$-boson systems, $B_{N+1}/B_N$, approaches a finite limit, approximately 8.567, at large $N$. We also confirm previous results that the three-body system has exactly two bound states. We find for the ground state $B_3^{(0)}=16.522688(1)B_2$ and for the excited state $B_3^{(1)}=1.2704091(1)B_2$.

Figure 1

Numerical solution of Eq. (17) for the boundary condition $f^{\prime }(0)=f(\infty )=0$ obtained using the shooting method.

Figure 2

The integral equation for the three-body amplitude. The single (double) line indicates the boson (full dimeron) propagator.